Among the achievements of Muslim mathematicians during this period include the development of algebra and algorithms by the Persian and Islamic mathematician Muhammad ibn Musa al-Khawarizmi the invention of spherical trigonometry, the addition of the decimal point notation to the Arabic numerals, the discovery of all the trigonometric functions besides sine, al-Kindi's introduction of cryptanalysis and frequency analysis, al-Karaji's introduction of algebraic calculus and proof by mathematical induction, the development of analytic geometry and the earliest general formula for infinitesimal and integral calculus by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam, the first refutations of Euclidean geometry and the parallel postulate by Nasir al-Din al-Tusi, the first attempt at a non-Euclidean geometry by Sadr al-Din, the development of symbolic algebra by Abu al-Hasan ibn Ali al-Qalasadi and numerous other advances in algebra, arithmetic, calculus, cryptography, geometry, number theory and trigonometry.
